Nearby theorems
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Nearby theorems |
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| Mirrors > Home > ILE Home > Th. List > lssvscl | GIF version | ||
| Description: Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
| Ref | Expression |
|---|---|
| lssvscl.f | β’ πΉ = (Scalarβπ) |
| lssvscl.t | β’ Β· = ( Β·π βπ) |
| lssvscl.b | β’ π΅ = (BaseβπΉ) |
| lssvscl.s | β’ π = (LSubSpβπ) |
| Ref | Expression |
|---|---|
| lssvscl | β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (π Β· π) β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpll 527 | . . 3 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β LMod) | |
| 2 | simprl 529 | . . . 4 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β π΅) | |
| 3 | simplr 528 | . . . . 5 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β π) | |
| 4 | simprr 531 | . . . . 5 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β π) | |
| 5 | eqid 2177 | . . . . . 6 β’ (Baseβπ) = (Baseβπ) | |
| 6 | lssvscl.s | . . . . . 6 β’ π = (LSubSpβπ) | |
| 7 | 5, 6 | lsselg 13454 | . . . . 5 β’ ((π β LMod β§ π β π β§ π β π) β π β (Baseβπ)) |
| 8 | 1, 3, 4, 7 | syl3anc 1238 | . . . 4 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β π β (Baseβπ)) |
| 9 | lssvscl.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
| 10 | lssvscl.t | . . . . 5 β’ Β· = ( Β·π βπ) | |
| 11 | lssvscl.b | . . . . 5 β’ π΅ = (BaseβπΉ) | |
| 12 | 5, 9, 10, 11 | lmodvscl 13401 | . . . 4 β’ ((π β LMod β§ π β π΅ β§ π β (Baseβπ)) β (π Β· π) β (Baseβπ)) |
| 13 | 1, 2, 8, 12 | syl3anc 1238 | . . 3 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (π Β· π) β (Baseβπ)) |
| 14 | eqid 2177 | . . . 4 β’ (+gβπ) = (+gβπ) | |
| 15 | eqid 2177 | . . . 4 β’ (0gβπ) = (0gβπ) | |
| 16 | 5, 14, 15 | lmod0vrid 13415 | . . 3 β’ ((π β LMod β§ (π Β· π) β (Baseβπ)) β ((π Β· π)(+gβπ)(0gβπ)) = (π Β· π)) |
| 17 | 1, 13, 16 | syl2anc 411 | . 2 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β ((π Β· π)(+gβπ)(0gβπ)) = (π Β· π)) |
| 18 | 15, 6 | lss0cl 13462 | . . . 4 β’ ((π β LMod β§ π β π) β (0gβπ) β π) |
| 19 | 18 | adantr 276 | . . 3 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (0gβπ) β π) |
| 20 | 9, 11, 14, 10, 6 | lssclg 13457 | . . 3 β’ ((π β LMod β§ π β π β§ (π β π΅ β§ π β π β§ (0gβπ) β π)) β ((π Β· π)(+gβπ)(0gβπ)) β π) |
| 21 | 1, 3, 2, 4, 19, 20 | syl113anc 1250 | . 2 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β ((π Β· π)(+gβπ)(0gβπ)) β π) |
| 22 | 17, 21 | eqeltrrd 2255 | 1 β’ (((π β LMod β§ π β π) β§ (π β π΅ β§ π β π)) β (π Β· π) β π) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 = wceq 1353 β wcel 2148 βcfv 5218 (class class class)co 5878 Basecbs 12465 +gcplusg 12539 Scalarcsca 12542 Β·π cvsca 12543 0gc0g 12711 LModclmod 13383 LSubSpclss 13448 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-pre-ltirr 7926 ax-pre-ltadd 7930 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-pnf 7997 df-mnf 7998 df-ltxr 8000 df-inn 8923 df-2 8981 df-3 8982 df-4 8983 df-5 8984 df-6 8985 df-ndx 12468 df-slot 12469 df-base 12471 df-sets 12472 df-plusg 12552 df-mulr 12553 df-sca 12555 df-vsca 12556 df-0g 12713 df-mgm 12781 df-sgrp 12814 df-mnd 12824 df-grp 12886 df-minusg 12887 df-sbg 12888 df-mgp 13137 df-ur 13149 df-ring 13187 df-lmod 13385 df-lssm 13449 |
| This theorem is referenced by: lssvnegcl 13469 islss3 13472 islss4 13475 lspsneli 13507 lspsn 13508 |
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